Separable Metric Space is Second-Countable
Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $M$ be separable.
Then $M$ is second-countable.
Proof
By the definition of separability, we can choose a subset $S \subseteq X$ that is countable and everywhere dense.
Define:
- $\BB = \set {\map {B_{1/n} } x: x \in S, \, n \in \N_{>0} }$
where $\map {B_\epsilon } x$ denotes the open $\epsilon$-ball of $x$ in $M$.
We have that Cartesian Product of Countable Sets is Countable.
Hence, by Image of Countable Set under Mapping is Countable, it follows that $\BB$ is countable.
Let $\tau$ denote the topology on $X$ induced by the metric $d$.
It suffices to show that $\BB$ is an analytic basis for $\tau$.
From Open Ball of Metric Space is Open Set, we have that $\BB \subseteq \tau$.
We use Equivalence of Definitions of Analytic Basis.
Let $y \in U \in \tau$.
By the definition of an open set, there exists a strictly positive real number $\epsilon$ such that $\map {B_\epsilon} y \subseteq U$.
By the Axiom of Archimedes, there exists a natural number $n > \dfrac 2 \epsilon$.
That is:
- $\dfrac 2 n < \epsilon$
and so:
- $\map {B_{2/n} } y \subseteq \map {B_\epsilon} y$.
From Subset Relation is Transitive, we have $\map {B_{2/n} } y \subseteq U$.
By the definition of everywhere denseness, and by Equivalence of Definitions of Adherent Point, there exists an $x \in S \cap \map {B_{1/n} } y$.
By Metric Space Axiom $(\text M 3)$, it follows that $y \in \map {B_{1/n} } x$.
For all $z \in \map {B_{1/n} } x$, we have:
\(\ds \map d {z, y}\) | \(\le\) | \(\ds \map d {z, x} + \map d {x, y}\) | Metric Space Axiom $(\text M 2)$: Triangle Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {z, x} + \map d {y, x}\) | Metric Space Axiom $(\text M 3)$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \frac 2 n\) |
That is:
- $\map {B_{1/n} } x \subseteq \map {B_{2/n} } y$
From Subset Relation is Transitive, we have:
- $y \in \map {B_{1/n} } x \subseteq U$
Hence the result.
$\blacksquare$
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis: Exercise $2.23$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 37$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces